So now we've introduced call options and put options. We've discussed long and short positions. And the links and differences between payoffs, profits, and intrinsic value. The next step is to identify the different factors that may impact upon the price paid for an option, which is what we call the option premium. Now it's important at this early stage to recognize that the value of an option consists of two separate components. Firstly, the intrinsic value of an option, which we discussed in the last session, is simply the payoff from the option if it were to expire immediately. So, for a call option, if the value of the underlying asset is less than the exercise price of the option. As is the case to the left-hand side of this graph, there is zero intrinsic value. For each dollar that the value of the asset exceeds the exercise price, the call option's intrinsic value also increases by $1. As indicated by the upward sloping straight line on the graph. Secondly, there is the time value of an option. The time value of an option is the difference between the total option value which is indicated by the curved line on the graph, and the option's intrinsic value. As we can see from this graph, an option still has value even when it is out of the money. As there is a chance in the remaining time before expiring that the prices may change and the option might provide a positive payoff. There are six factors that affect the value of an option. As each of these factors change, we will document the impact of that change on the value of the option, and highlight which component of the option's value, intrinsic or time value, upon which the change has the greatest effect. In this type of analysis, it's very useful to change one variable at a time, and assume that all the other variables are held constant. So the first variable to consider is the option's exercise price. This one's simple. A call option gives you the right to pay for an asset in the future. The right to pay a higher price is worth less than the right to pay a lower price. And hence, we have a negative relationship between call option values and exercise prices. Conversely, a put option providers the holder with the right to receive the exercise price. And hence, the right to receive a higher price is worth more than the right to receive a lower price. And hence, we have a positive relationship between put values and exercise prices. Holding exercise price constant now. We know that the value of the call option increases with the value of the underlying asset. So we have a positive relationship between calls and asset values. Naturally, the value of the put option declines as the value of the underlying asset increases as the moniness of the put option also declines. Interest rates impact upon option values primarily via the discounting process. Recall that a call option involves potentially paying the exercise price at a future date. The present value of this cash outflow will decline as interest rates increase. And hence we see a positive relationship between option values and interest rates. As a put option holder receives the exercise price at expiry, if the option is exercised, and the present value of that cash inflow decreases as interest rates increase, we see a negative relationship between put option values and interest rates. Now hopefully, you recall from module two of this course, the impact that the dividends had on share prices. Specifically on the ex-dividend date, the value of a share is expected to decline by the approximate value of the dividend that had been declared. So dividends act to reduce the value of the underlying asset, which in our case, the present case, is a share. Well, we've already discussed that a decrease in the value of the underlying asset will decrease the value of the call option. So there is a negative relationship between the size of the dividend and the value of a call option while the relationship is reverse for put options. Now, while we're discussing dividends, let's pause for a moment to talk about the ability to exercise an option, before its expiry date. Recall, that while European-style options can only be exercised at expiry. American-style options can be exercised at any time, up to and including the expiry date. When you own an American-style call option, on the upside it's just like holding the underlying asset in that you benefit from increases in the value of the asset beyond the exercise price, but without the downside associated with asset price declines as your losses there are capped at the premium you paid for the option. So, in most cases, it never really makes sense to exercise an American-style call option early. Now, one exception to that rule is where the asset is about to go ex-dividend. Because the call option holder is entitled to the dividend and the asset price will fall on the ex-dividend date, it may be optimal for the option holder to exercise the call option early so that they can take delivery of the asset and pocket the dividend. So what about put options? Unlike call options, put options involve possibly receiving the exercise price in the future. That implies that the holder of a put option is delaying receipt of the cashflow, by choosing not to exercise early. It may make sense to exercise a put option early if it is deep in the money. So deep that you are very unlikely to regret selling at the exercise price between now and when the option would have expired. Whereas the previous fall variables all impacted upon the intrinsic value of the option, the next two have an impact upon the time value of the option. The first factor is volatility in prices. Now, to demonstrate this, let's assume the following. The current price of corn is $7.50. We're considering options that expire on 31st of March, 2014, and the options we are considering are a call option held by Judy with an exercise price of $7.50, and a put option held by Frank, also with an exercise price of $7.50. Note that in this example, both Judy and Frank have options that are currently at the money. To demonstrate the impact of volatility on option prices, we are going to run 6 price simulations where using some fancy modeling, we simulate what might happen to the price of corn over the next three months. Now a key input into the simulation is volatility, which is simply a measure of how variable prices are in the market for corn. In our first set of simulations, let's assume a low level of volatility. Each of the lines on this graph represents a different simulated price series. The next step is to calculate the payoff to each of Judy and Frank's options by comparing the final simulated price on 31st of March with the exercise price of their options. So the first simulated price here is finishes on a price of $8.01. So Judy's payoff for her call option with an exercise price of $7.50 is $0.51. Frank's put option with an exercise price of $7.50 has finished out of the money. So here's a payoff of $0 from the first simulated price series. We continue with this process, recording payoffs to each option for each simulated series. At the end, we average the payoffs across the sample and that gives us an estimate of the average payoff from the option. So, in this case, our estimates are $0.38 for the call option and $0.14 for the put option. Now let me pause here. In reality, this is actually the approach we often take to value derivative securities. Although, normally, we might generate, say, 100,000 simulated price series, to make sure we have as accurate an estimate as possible of the option's value. Now let's re-run the simulations, but this time let's increase the assumed level of of volatility in the price of corn. As expected, prices are a lot more variable. Indeed, what we notice is that the simulator price of corn on the 31st of March is likely to finish a much greater distance from its starting value. When we estimate the average payoff of duty and Frank's options, we find that both the average payoff to the call and the average payoff to the put are both higher than before. Now the reason for this is that options provide asymmetric payoffs. That is they benefit more from beneficial price movements than they suffer from adverse price movements. For example, consider simulated series one. When volatility was low, the final simulated price was $8.01. And the payoff to the call was $0.51, and the payoff to the put was $0. Now, with high volatility, the final simulated price is $8.65. The payoff to the call is much higher at $1.15. Yet, the payoff to the put, is still $0, just as it was before. Similarly, when simulated prices fall below the exercise price of the put option, as we simulated series three, we find that it falls far below, providing a payoff of $1.13, rather than a measly $0.27, which was the payoff, in a low volatility environment. In short, volatility provides a net benefit to all option holders, as it increases the chance that the option will finish deeper in the money. Now, true, it also increases the chance that the option will expire deeper out of the money. But that doesn't bother the option holder, as the payoff to finishing a little out of the money, or deep out of the money, is identical in both cases. You net at $0. The final factor to consider is time to expiry for an option. Generally speaking, this is very straightforward as it is analogous to the situation we face with volatility as the time to expiry for an option increases, so to does the chance that the option will finish even deeper in the money. In summary, we started the session by decomposing option values into their constituent parts, intrinsic value, and time value. We highlighted that even where an option had no intrinsic value, where it was currently out of the money, it still had a positive time value. We then discussed the six factors that impact upon both call and put options. Highlighting how exercise prices, asset values, interest rates and dividends primarily affected option values via adjustments to intrinsic value. While price volatility and time to expiry both impact upon the time value of the options. In our next session together, we're going to demonstrate some pretty cool ways in which options might be combined to provide some quite interesting payoffs.
